Robust pricing--hedging duality for American options in discrete time financial markets

We investigate pricing-hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, e.g. a family of European options, only statically. In the first part of the paper we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a non-dominated family of probability measures. Our first insight is that by considering a (universal) enlargement of the space, we can see American options as European options and recover the pricing-hedging duality, which may fail in the original formulation. This may be seen as a weak formulation of the original problem. Our second insight is that lack of duality is caused by the lack of dynamic consistency and hence a different enlargement with dynamic consistency is sufficient to recover duality: it is enough to consider (fictitious) extensions of the market in which all the assets are traded dynamically. In the second part of the paper we study two important examples of robust framework: the setup of Bouchard and Nutz (2015) and the martingale optimal transport setup of Beiglb\"ock et al. (2013), and show that our general results apply in both cases and allow us to obtain pricing-hedging duality for American options.


INTRODUCTION
The robust approach to pricing and hedging has been an active field of research in mathematical finance over recent years. It aims to address one of the key shortcomings of the classical approach, namely, its inability to account for model misspecification risk. Accordingly, the capacity to account for model uncertainty is at the core of the robust approach. In the classical approach, one postulates a fixed probability measure ℙ to describe the future evolution of prices of risky assets. In contrast, in the robust approach, one considers the pricing and hedging problem simultaneously under a family of probability measures, or pathwise on a set of feasible trajectories. The challenge lies in extending the arbitrage pricing theory, which is well understood in the classical setup, to the robust setting.
In the classical approach, when the reference measure ℙ is fixed, the absence of arbitrage is equivalent to the existence of a martingale measure equivalent to ℙ, a result known as the first fundamental theorem of asset pricing, see, for example, Delbaen and Schachermayer (2006) or Föllmer and Schied (2004). When the market is complete-that is, when every contingent claim can be perfectly replicated using a self-financing trading strategy-the equivalent martingale measure ℚ is unique, and the fair price for a contingent claim is equal to the replication cost of its payoff, and may be computed as the expected value of the discounted payoff under ℚ. In an incomplete market, where a perfect replication strategy does not always exist, a conservative way of pricing is to use the minimum superreplication cost of the option. Employing duality techniques, this superreplication price can be expressed as the supremum of expectations of the discounted payoff over all martingale measures equivalent to ℙ.
In the robust approach, in the absence of a dominating probability measure, this elegant story often becomes more involved and technical. In continuous time models under volatility uncertainty, analogous pricing-hedging duality results have been obtained by, among many others, Denis and Martini (2006), Soner, Touzi, and Zhang (2013), Neufeld and Nutz (2013), and Possamaï, Royer, and Touzi (2013). In discrete time, a general pricing-hedging duality was shown in, for example, Bouchard and Nutz (2015) and Burzoni, Frittelli, and Maggis (2017). Importantly, in a robust setting, one often wants to include additional market instruments, which may be available for trading. In a setup that goes back to the seminal work of Hobson (1998), one often considers dynamic trading in the underlying asset and static trading, that is, buy and hold strategies at time zero, in some European options, often call or put options with a fixed maturity. Naturally, such additional assets constrain the set of martingale measures, which may be used for pricing. General pricing-hedging duality results, in different variations of this setting, both in continuous and in discrete time, can be found in, for example, Acciaio, Beiglböck, Penkner, and Schachermayer (2016), Burzoni, Frittelli, Hou, Maggis, and Obłój (2018), Beiglböck et al. (2013), Dolinsky and Soner (2014), Hou and Obłój (2018), Guo, Tan, and Touzi (2017), Tan and Touzi (2013), and we refer to the survey papers Hobson (2011) and Obłój (2004) for more details.

PRICING-HEDGING DUALIT Y FOR AMERICAN OPTIONS
We present in this section general results that explain when and why the pricing-hedging duality for American options holds. We work in a general discrete time setup, which we now introduce. Let (Ω,  ) be a measurable space and ∶= ( ) =0,1,…, be a filtration, where  0 is trivial and ∈ ℕ is the time horizon. We denote by (Ω) the set of all probability measures on (Ω,  ) and consider a subset  ⊂ (Ω). We say that a given property holds -quasi surely (-q.s.) if it holds ℙ-almost surely for every ℙ ∈ , and say that a set from  is -polar if it is a null set with respect to every ℙ ∈ . We write ℚ ⋘  if there exists a ℙ ∈  such that ℚ ≪ ℙ. Given a random variable and a sub--field  ⊂  , we define the conditional expectation ℙ [ |] ∶= ℙ [ + |] − ℙ [ − |] with the convention ∞ − ∞ = −∞, where + ∶= ∨ 0 and − ∶= −( ∧ 0). We consider a market with no transaction costs and with financial assets, some of which are dynamically traded and some of which are only statically traded. The former are modeled by an adapted ℝ -valued process with ∈ ℕ. We think of the latter as European options, which are traded at time = 0 and not at future times. We let = ( ) ∈Λ , where Λ is a set of arbitrary cardinality, be the vector of their payoffs, which are assumed to be ℝ-valued and  -measurable. Up to a constant shift of the payoffs, we may assume, without loss of generality, that all options have zero initial price. All prices are expressed in units of some numeraire 0 , such as a bank account, whose price is thus normalized, so that 0 ≡ 1. We denote by  the set of allpredictable ℝ -valued processes, and by = {ℎ ∈ ℝ Λ ∶ ∃ finite subset ⊂ Λ s.t. ℎ = 0∀ ∉ }. A self-financing strategy trades dynamically in and statically in finitely many of , ∈ Λ and hence corresponds to a choice of ∈  and ℎ ∈ . Its associated final payoff is given by where Δ = − −1 . Having defined the trading strategy, we can consider the superhedging price of an option with payoff at time , given by In particular, if  = (Ω) is the set of all probability measures on  and { } ∈  for all ∈ Ω, then the superreplication in (2) is pathwise on Ω.
To formulate a duality relationship, we need the dual elements given by rational pricing rules, or martingale measures, Definition 2.1. Let Υ be a given class of real-valued functions defined on Ω. We say that the (European) pricing-hedging duality holds for the class Υ if  ≠ ∅ and Remark 2.2. Note that the inequality "≥" in (4), called weak pricing-hedging duality, holds automatically from the definition of  in (3).
A number of papers, including Bouchard and Nutz (2015) and Burzoni et al. (2018), proved that the above pricing-hedging duality (4) holds under various further specifications and restrictions on Ω, ,  and Υ, including in particular an appropriate no-arbitrage condition. We take the above duality for granted here and our aim is to study an analogous duality for American options. We work first in the general setup described above without specifying or Υ, as our results will apply to any such further specification. Further, many abstract results in this section also extend to other setups, for example, to trading in continuous time.

Superhedging of American options
An American option may be exercised at any time ∈ ∶= {1, … , } (without loss of generality we exclude exercise at time 0). It is described by its payoff function Φ = (Φ ) 1≤ ≤ , where Φ ∶ Ω → ℝ belongs to Υ and is the payoff, delivered at time , if the option is exercised at time . Usually Φ is taken to be  -measurable, but here we only assume Φ to be  -measurable for greater generality, which includes, for example, the case of a portfolio containing a mixture of American and European options. We note that when hedging our exposure to an American option, we should be allowed to adjust our strategy in response to an early exercise. As a consequence, the superhedging cost of the American option Φ using semistatic strategies is given by Remark 2.3. We formulate the problem above with payoff delivered by the seller at maturity irrespectively of the actual exercise time. In full generality, this is necessary because the payoff is not assumed to be known at the exercise time. However, given that we work in discounted units, if the payoff is known at the exercise time, our convention is equivalent to the one in which the payoff is delivered at its exercise time, via taking a loan, and then the seller has to be able to continue trading in such a way that her final payoff is nonnegative. For this equivalence to hold it is important to allow the seller to adjust the strategy at the time of the exercise. Note that in the more classical setting when Φ is measurable and there are no statically traded options, that is, Λ = ∅, no-arbitrage ensures that to have a final nonnegative payoff the seller has to have a nonnegative wealth after delivering the payoff at the exercise time. She can then just stop trading altogether-the vector of strategies ( 1 , … , ) ∈  above then reduces to a single trading strategy which is unwound at the exercise time.
Classically, the pricing of an American option is recast as an optimal stopping problem and, extending (4), it would be natural to ask whether holds, where  ( ) denotes the set of -stopping times. However, as illustrated by the simple example in the introduction, this duality may fail. The "numerical" reason is that the right-hand side in (4) may be too small because the set  ×  ( ) is too small. Our aim here is to understand the fundamental reasons why the duality fails and hence discuss how and why the right-hand side should be modified to obtain the equality in (5).

An American option is a European option on an enlarged space
The first key idea of this paper offers a generic enlargement of the underlying probability space which turns all American options into European options. Depending on the particular setup, it may take more or less effort to establish (4) for the enlarged space, but this shifts the difficulty back to the better understood and well studied case of European options. Our reformulation technique-from an American to European option-can be easily extended to other contexts, such as the continuous time case. The enlargement of space is based on construction of random times, previously used, for example, in Song (2011a, 2011b) to study the existence of random times with a given survival probability, in El Karoui and  to study a general optimal control/stopping problem, and in Guo, Tan, and Touzi (2016) and Källblad, Tan, and Touzi (2017) to study the optimal Skorokhod embedding problem.
Recalling the notation ∶= {1, … , }, we introduce the space Ω ∶= Ω × with the canonical time ∶ Ω → given by ( ) ∶= , where ∶= ( , ), the filtration ∶= ( ) =0,1,…, with  =  ⊗ and = ( ∧ ( + 1)), and the -field  =  ⊗ . By definition, is an -stopping time. We denote by  the class of -predictable processes and extend naturally the definitions of and from Ω to Ω via ( ) = ( ) and ( ) = ( ) for = ( , ) ∈ Ω. We let Υ be the class of random variables ∶ Ω → ℝ such that (⋅, ) ∈ Υ for all ∈ and we let (̄) denote the superreplication cost of . We may, and will, identify Υ with Υ via ( ) = Φ ( ). Finally, we introduce Theorem 2.4. For any Φ ∈ Υ = Υ, we have In particular, if the European pricing-hedging duality on Ω holds for Φ, then Proof. First note that and hence that the dynamic strategies used for superhedging in and in are the same. The equality now follows by observing that a set Γ ∈  is -polar if and only if its -sections Γ = { ∶ ( , ) ∈ Γ} are -polar for all ∈ . Indeed, for one implication assume that ℙ(Γ) = 0 for each ℙ ∈ . For arbitrary ℙ ∈  and ∈ we can define ℙ = ℙ ⊗ , which belongs to , and hence ℙ(Γ ) = 0 follows. To show the reverse implication, assume that ℙ(Γ ) = 0 for each ℙ ∈  and ∈ . Observe that, for any ℙ ∈ , as ℙ |Ω ∈ . This completes the proof. □ Remark 2.5. If the pricing-hedging duality holds with respect to the filtration , then it also holds for any filtration ℍ ⊃ such that ℍ and only differ up to  -polar sets. Indeed, this follows from Remark 2.2, observing that such a change does not affect  and can only decrease the superhedging cost as one has more trading strategies available.
Remark 2.6. We note that the set  in (8) is potentially much larger than the set of all pushforward measures induced by  → ( , ( )) and ℚ ∈  for all ∈  ( ). Indeed, instead of stopping times relative to , it allows us to consider any random time, which can be made into a stopping time under some calibrated martingale measure. We can rephrase this as saying that  is equivalent to a weak formulation of the initial problem on the right-hand side of (5). To make this precise, let us define a weak stopping rule as a collection with (Ω ,  , ℚ , ) a filtered probability space, a -valued -stopping time, an ℝ -valued (ℚ , )-martingale , and a collection of random variables , , Φ , and such that there is a measurable surjective mapping ∶ Ω → Ω with ℚ = ℚ • −1 ∈  and −1 ( ) ⊂  , −1 ( ) ⊂  , and finally  ℚ ( , , Φ ) =  ℚ ( , , Φ). Denote by  the collection of all weak stopping rules such that ℚ [ , ] = 0 for each ∈ Λ. It follows that any ∈  induces a probability measure ℚ ∈  and ℚ [Φ ] = ℚ [Φ]. Reciprocally, any ℚ ∈  , together with the space (Ω,  , ) and ( , , Φ), provides a weak stopping rule in  . As a consequence, In summary, and similarly to a number of other contexts, see the introduction in Pham and Zhang (2014), the weak formulation (and not the strong one) offers the right framework to compute the value of the problem. In fact, the set  is large enough to make the problem static, or European, again. However, although it offers a solution and a corrected version of (5), it does not offer a fundamental insight into why (5) may fail and if there is a canonical "smaller" way of enlarging the objects on the right-hand side thereof to preserve the equality. These questions are addressed in the subsequent section.
Remark 2.7. Neuberger (2007) and Hobson and Neuberger (2017) studied the same superhedging problem in a Markovian setting, where the underlying process takes values in a discrete lattice . By considering the weak formulation (which is equivalent to our formulation, as shown in Remark 2.6 above), they obtain similar duality results. However, they only consider Φ = ( ), where ∶ ℝ → ℝ. Then, the authors show that in the optimization problem sup ℚ∈ ℚ [Φ] given in (7) one may restrict to only Markovian martingale measures. The primal and the dual problem then turn out to be linear programming problems under linear constraints, which can be solved numerically. Their arguments have also been extended to a more general context, where takes values in ℝ + . Comparing to Neuberger (2007) and Hobson and Neuberger (2017), our weak formulation is very similar to theirs. However, our setting is much more general and, when considering the specific setups in Sections 3 and 4, we rely on entirely different arguments to prove the duality.

The loss and recovery of the dynamic programming principle and the natural duality for American options
The classical pricing of American options, on which the duality in (5) was modeled, relies on optimal stopping techniques, which subsume a certain dynamic consistency, or a dynamic programming principle, as explained below. Our second key observation in this paper is that if the pricing-hedging duality (5) for American options fails it is because the introduction of static trading of European options at time = 0 destroys the dynamic programming principle. Indeed, ( ) will typically be lower than the superhedging price at time = 0 of the capital needed at time = 1 to superhedge from thereon. To reinstate such dynamic consistency, we need to enlarge the model and consider dynamic trading in options in . This will generate a richer filtration than and one which will carry enough stopping times to obtain the correct natural duality in the spirit of (5). In particular, if = 0 (or equivalently Λ = ∅), then (5) should hold. We now first prove this statement and then present the necessary extension when is nontrivial. Let Υ be a class of  -measurable random variables such that −∞ ∈ Υ, we denote ( ) ∶= sup ℚ∈ ℚ [ ], and suppose that there is a family of operators Notice that  0 is assumed to be trivial so that  0 ( ) is deterministic. We say that the family ( ) provides a dynamic programming representation of  if The family ( ) naturally extends to ( ), 0 ≤ ≤ − 1, defined for any Φ ∈ Υ = Υ by Assume that ∨ ′ ∈ Υ for , ′ ∈ Υ so that ( ) maps functionals from Υ to Υ. We introduce the following process: which, under suitable assumptions (see Proposition 2.9 below) represents the -Snell envelope process of an American option Φ ∈ Υ. To illustrate how the operator  0 works, we develop it for the case = {1, 2, 3}, ) .
We say that the family ( ) provides a dynamic programming representation of Typically we will consider  to be a supremum over conditional expectations with respect to  (see Examples 2.11 and 2.12 below), and in such setups we automatically obtain Theorem 2.8. Assume that Λ = ∅,  satisfies (9), that (13) holds true, and that ∨ ′ ∈ Υ for all , ′ ∈ Υ. Then, for all Φ ∈ Υ = Υ, If, further, the European pricing-hedging duality holds on Ω for the class Υ, then The second assertion follows instantly from the first one and Theorem 2.4. The first assertion is reformulated and proved in Proposition 2.9 below, which also allows us to identify the optimal stopping time on the right-hand side of (14). Proposition 2.9. Assume that Λ = ∅ and ∨ ′ ∈ Υ for all , ′ ∈ Υ. Then the dynamic programming representation (12) holds if and only if (9) and (13) hold true. Moreover,under condition (12), provides the optimal exercise policy for Φ ∈ Υ, in sense that Remark 2.10. The proof of Proposition 2.9 will be provided in Section 5. The results in Theorem 2.8 and Proposition 2.9 are stated on (Ω,  ), where there are only finitely many dynamically traded risky assets. However, their proofs do not rely on the fact that the number of risky assets is finite, and the same results still hold true if there are infinitely many dynamically traded risky assets.
Example 2.11. The model-specific setting is recovered by taking  = {ℙ}, for a fixed probability measure ℙ. Then, taking Υ to be the set of all  -measurable random variables and where the essential supremum is taken with respect to ℙ, leads to a family of operators satisfying (9), (13), and therefore also (12). See the literature on dynamic coherent risk measures for further discussion, for example, Acciaio and Penner (2011) for an overview. In particular, Theorem 2.8 recovers the classical superhedging theorem for American options (see, e.g., Myneni, 1992).
Example 2.12. Let (Ω, ) be a Polish space,  the universally completed Borel -field,  a given set of probability measures on (Ω,  ), and  be defined by (3). We are given a filtration ∶= ( ) ≤ such that  0 = {∅, Ω} and each -field  is countably generated. Let  be the universal completion s. for all  -measurable and ℙ ∈ , and the fact that  is countably generated ensures the existence of a regular conditional probability of ℙ with respect to  . Assume there exists a family ( ( )) ≤ −1, ∈Ω of sets of measures satisfying: Note that [ ]  ∈  because the latter is countably generated.
Let us consider the case with statically traded options: Λ ≠ ∅. We saw in Example 1.1 that this can break down dynamic consistency as the universe of traded assets differs at time = 0 and times ≥ 1.
To remedy this, one has to embed the market into a fictitious larger one where both and all the options , ∈ Λ, are traded dynamically.
Remark 2.14. A measure ℚ ∈  is an admissible pricing measure under which the time-prices for European options are given by ,ℚ . Property 4 in the above definition says that any such ℚ ∈  can be lifted to a measure (ℚ) ∈, which preserves the joint distribution of the stock and option prices. In general, we do not expect the reverse to be true and we may have ( ) ⊊. More precisely, may offer scope for a richer description and dynamics so that the mapping ∋Q  → ℚ• −1 ∈  is surjective but typically not injective.
Example 2.15. In practice, the map in a dynamic extension is often built from a family of mappings from Ω toΩ. Assume that, for each ℚ ∈  , one has a mapping ℚ ∶ Ω →Ω such that Then, ∶  → satisfies Property 4 of Definition 2.13.
Remark 2.16. Clearlŷis much richer than as it captures not only the evolution of prices of but also of all the vanilla options. The inequalitŷ(Φ) ≤ (Φ) holds trivially as a buy-and-hold strategy is a special case of a dynamic trading strategy and  =• −1 .
The following result shows that if the pricing-hedging duality holds then the superhedging prices in the fictitious dynamic extension market are the same as in the original market. This will apply to the setups we consider in Sections 3 and 4 below.
(a) Assume that the European pricing-hedging duality holds for the class Υ on Ω. Then (b) Assume that the European pricing-hedging duality holds for the class Υ on Ω. Then Proof. Note that ≥̂holds by Remark 2.16. Using (7) twice we obtain where the penultimate inequality always holds by Remark 2.2 and the last inequality follows by (20). The assumed pricing-hedging duality on Ω implies that we have equalities throughout. The proof of (a) is analogous but simpler. □ Remark 2.18. The above result may at first seem surprising. The dynamic extension introduces many new dynamically traded assets, yet the superhedging prices remain the same. The intuition behind this is that under pricing-hedging duality, the cheapest superhedge is a perfect hedge (or very nearly so) under some (worst case) model. Our dynamic extensions do not introduce any constraints on the prices of options and hence do not restrict the set of martingale measures. The worst case model will remain an admissible model and for this model the additional traded assets make no difference. They could however make a difference in many other (specific) models. If we considered a restricted version of dynamic trading in which we make further assumptions about the price dynamics of vanilla options, then this could imply that • −1 is not surjective and the superhedging prices might strictly decrease. Such a setup is studied in Nadtochiy and Obłój (2017), where the authors consider restrictions on the levels of implied volatility through time.
Let (Ω,̂, ,, , , ) be a dynamic extension of (Ω, ,  , , , ) and be a family of operators on the spaceΥ of functionals onΩ. One can define the corresponding extended operators as well as as in (9) and (10). We can then apply Theorem 2.8 and Proposition 2.17 to obtain the following result: Corollary 2.19. Let (Ω,̂, ,, , , ) be a dynamic extension of (Ω, ,  , , , ) with operatorŝ  ∶Υ →Υ and the corresponding extended operators as well as satisfying (9) and (13). Assume that the European pricing-hedging duality holds for the class Υ on Ω, and ∨ ′ ∈ Υ for all , ′ ∈ Υ. Then, for all Φ ∈ Υ , (2, 2/5 ) Remark 2.20. In Section 3, in the context of Bouchard and Nutz (2015), we will adopt the dynamic extension introduced in Example 2.15, and show that it admits a family of operators to which we can apply Corollary 2.19.
Remark 2.21. We believe that Corollary 2.19 describes a canonical, and in some sense minimal, solution to the pricing-hedging duality of American option, when compared to addition of all consistent random times, as discussed in Remark 2.6. A dynamic extensionΩ is crucial to establish the Dynamic Programming Principle (DPP), which in turn allows one to define the optimal stopping timê * .
Remark 2.22. Let us consider the two period ( = 2) example of Hobson and Neuberger (2016); see Figure 2. For simplicity, we introduce only one statically traded option with payoff 1 1 { 2 =4} at time = 2 and price 2∕5 at time = 0. This already destroys the pricing-hedging duality for the American option Φ. In Hobson and Neuberger (2016), the duality is recovered by considering a (calibrated) mixture of martingale measures. It is insightful to observe that their mixture model is nothing else but a martingale measure for an augmented setup with dynamic trading in , which, following Corollary 2.19, restores the dynamic programming principle and the pricing-hedging duality for American options. To show this, let denote the price process of the option , so that 0 = 2∕5 and 2 = . Figure 2 illustrates a martingale measure ℚ along with the intermediate prices 1 such that the processes and are martingales. With = 1 1 { 1 =1, 1 =0} + 21 1 { 1 =1, 1 =1∕4}∪{ 1 =3} , we find ℚ [Φ ] = 18∕5, which is the superhedging price, and the duality is recovered.

Pseudo-stopping times
In this subsection we study the connection of our problem to pseudo-stopping times in the filtration which form a bigger class than -stopping times. We refer the reader to Williams (2002), Nikeghbali and Yor (2005) and Mansuy and Yor (2006) for an introduction to pseudo-stopping times.
It follows from Theorem 2.4 that in general we expect to see where the inequality may be strict. We showed above that this is linked with the necessity to use random times beyond ∈  ( ). To conclude our general results, we explore this property from another angle and identify the subset(s) of  , which lead to equality in the place of the inequality above. We introduce as the set of measures that make an -martingale and an -pseudo-stopping time. These are natural because the martingale part of the Snell envelope can be stopped at the pseudo-stopping time with null expectation.
Proposition 2.23. Assume that  ≠ ∅. Then Proof. Let ℚ ∈  such that ℚ [| |] < ∞ and ℚ [|Φ |] < ∞ for all ∈ Λ and = 1, … , . We next consider the optimal stopping problem sup ∈ ( ) which is an ( , ℚ)-supermartingale. Its Doob-Meyer decomposition is given by ≤ is an -predictable increasing process, and 0 = 0 = 0. It follows that We deduce that sup ℚ∈ . Then, (26) holds as every stopping time ∈  ( ) is a pseudo-stopping time and hence the inverse inequality is trivial. □ Remark 2.24. The above allows us to see that it is not enough to use randomized stopping times to recover the equality in (5). Such a time corresponds to an -adapted increasing process with 0 = 0 and = 1. It may be seen as a distribution over all possible stopping times, in our setup a distribution on such that ({ }) ∶= Δ = − −1 for each ∈ . For any pseudo-stopping time , the dual optional projection of the process 1 1 [[ , ]] is a randomized stopping time. Conversely, for a given , if we take a uniformly distributed random variable Θ independent of , possibly enlarging the probability space, then ∶= inf { ∶ ≥ Θ} is -pseudo-stopping time, which generates . Let  be the set of such randomized stopping times. Then, from Proposition 2.23 and the definition of the dual optional projection, Remark 2.25. Nikeghbali and Yor (2005) showed that under a progressive enlargement with pseudostopping time , all martingales from the smaller filtration stopped at remain martingales in the larger filtration. One can relate this to a more restrictive situation, when all martingales from the smaller filtration remain martingales in the bigger filtration, which is called the immersion property in the context of filtration enlargement. Clearly each random time satisfying the immersion property is a pseudo-stopping time. Thus, keeping the equality (26) true, the pseudo-stopping time property in the definition of  above can be replaced by a stronger condition characterizing the immersion property, See section 3.1.2 of Blanchet-Scalliet, Jeanblanc, and Romero (2016) for the discrete time context of progressive enlargement of filtration and Aksamit and Li (2016) for connections between pseudostopping times, the immersion property, and projections.

A DETAILED STUDY OF THE NONDOMINATED SETUP OF BOUCHARD AND NUTZ (2015)
In this section we work in the nondominated setup introduced in Bouchard and Nutz (2015), which is a special case of Example 2.12. We let Ω 0 = { 0 } be a singleton and Ω 1 be a Polish space. For each ∈ {1, … , }, we define Ω ∶= { 0 } × Ω 1 as the -fold Cartesian product. For each , we denote  ∶= (Ω ) and by  its universal completion. In particular,  0 and  0 are trivial, and ℙ [ | ] = ℙ [ | ] for all ∈  and every probability measure ℙ on (Ω ,  ). We shall often see  and  as sub--fields of  , and hence obtain two filtrations = ( ) 0≤ ≤ and = ( ) 0≤ ≤ on Ω. Denote Recall that a subset of a Polish space Ω is analytic if it is the image of a Borel subset of another Polish space under a Borel measurable mapping. We take Υ to be the class of upper semianalytic functions The price process is a -adapted ℝ -valued process and the collection of options = ( 1 , … , ) is a -measurable ℝ -valued vector for ∈ ℕ (thus Λ = {1, … , }).
Let ∈ {0, … , − 1} and ∈ Ω . We are given a nonempty convex set  ( ) ⊆ (Ω 1 ) of probability measures, which represents the set of all possible models for the ( + 1)th period, given state at times 0, 1, … , . We assume that for each , Given a universally measurable kernel ℙ ∶ Ω → (Ω 1 ) for each ∈ {0, 1, … , − 1}, we define a probability measure ℙ We can then introduce the set  ⊆ (Ω) of possible models for the multiperiod market up to time by Notice that the condition (29) ensures that  always has a universally measurable selector: ℙ ∶ Ω → (Ω 1 ) such that ℙ ( ) ∈  ( ) for all ∈ Ω . Then the set  defined in (30) is nonempty. We also denote where ⊗ ℚ ∶= ( 1 ,…, ) ⊗ ℚ is a Borel probability measure on Ω +1 ∶= Ω × Ω 1 , and The following notion of no-arbitrage NA() has been introduced in Bouchard and Nutz (2015): Analogously, we will say that NA() holds if for all ( , ℎ) ∈  × ℝ , Recall also that  and  have been defined in (3) and (6). As established in Bouchard and Nutz (2015), the condition NA() is equivalent to the statement that  and  have the same polar sets.
The following lemma extends this result to Ω. Proof. The two conditions NA() and NA() are equivalent by the same arguments as in proving (7). It is enough to show that  and  have the same polar sets if and only if  and  have the same polar sets. This boils down to proving that a set Γ ∈ Ω is an  polar set if and only if the -section Γ = { ∶ ( , ) ∈ Γ} is an  polar set for each ∈ , which can be shown similarly to the analogous statement involving  and  established in the proof of Theorem 2.4. □

Duality on the enlarged space
Our first main result is the following duality under the no-arbitrage condition (33): Theorem 3.2. Let NA() hold. Then the set  is nonempty, and, for any upper semianalytic Φ ∶ Ω → ℝ, one has and in particular the pricing-hedging duality (8) holds. Moreover, there exists ( , ℎ) ∈  × ℝ such that The proof is postponed to Section 6 and uses the following lemma. Let us work with the operators  introduced in Example 2.12 with  ( ) defined as in (32). Observe that By Proposition 2.9, (4.12) in Bouchard and Nutz (2015) and using that the maximum of upper semianalytic functions is still upper semianalytic, we conclude the following: (10) is also upper semianalytic and

(a)
The above result is in fact a classical dynamic programming principle result studied in Bertsekas and Shreve (2007) and Dellacherie (1985). The only crucial step is to prove that the graph set (b) Assume that NA() holds. It then follows by Lemma 3.1 and Theorem 3.2 that the pricing-hedging duality on Ω in (34) holds. Further, by defining and with as in (9) and (11), one has that (9) holds for from Theorem 3.4, and moreover that (13) holds for as it is a special case of Example 2.12. It then follows by Corollary 2.19 that (24) holds in this framework.
More recently, Bayraktar and Zhou (2017) consider the "randomized" stopping times, and obtain a more complete duality for (Φ). The dual formulations in Bayraktar and Zhou (2017) and in our results are more or less in the same spirit, as in Neuberger (2007) and Hobson and Neuberger (2017). Nevertheless, the duality in Bayraktar and Zhou (2017) is established under strong integrability conditions and an abstract condition, which is checked under regularity conditions (see their assumption 2.1 and remark 2.1). In particular, when  is the class of all probability measures on Ω, the integrability condition in their assumption 2.1 is equivalent to saying that Φ and are all uniformly bounded. In our paper, we only assume that are Borel measurable, Φ are upper semianalytic and all are ℝ-valued.
Technically, Bayraktar and Zhou (2017) use the duality results in Bouchard and Nutz (2015) together with a minimax theorem to prove their results. Our first main result consists of introducing an enlarged canonical space (together with an enlarged canonical filtration) to reformulate the main problem as a superhedging problem for European options. Then, by adapting the arguments in Bouchard and Nutz (2015), we establish our duality under general conditions as in Bouchard and Nutz (2015). Moreover, we do not assume that Φ is  -measurable, which enables us to study the superhedging problem for a portfolio containing an American option and some European options. Finally, our setting enables us to apply an approximation argument to study a new class of martingale optimal transport problems and to obtain a Kantorovich duality as in Section 4.
The condition (37) ensures the existence of a calibrated martingale measure, that is, that the following sets are nonempty: Recall that  =  . As Λ is a linear space, the superhedging cost of the American option Φ using semistatic strategies (Φ) defined in Section 2.1 can be rewritten as Similarly, we denote by (Φ) the corresponding superhedging cost for a European option with payoff Φ defined on Ω, and one has (Φ) = (Φ) by Theorem 2.4. Let us now construct a martingale measure ℚ 0 by Then, one can check that ℚ 0 ∈  , and it follows that The superhedging price of Φ is equal to 3∕2, as one can consider a superhedging strategy consisting of holding 3∕2 in cash and one option from Example 1.1. In a similar way as in Example 1.1, the duality may be recovered by allowing dynamic trading options.

Duality on the enlarged space
The following theorem shows the duality for Ω. Its proof is postponed to Section 7.
Theorem 4.2. Suppose that Φ ∶ Ω → ℝ is bounded from above and upper semicontinuous. Then, there exists an optimal martingale measure ℚ * ∈  and the pricing-hedging duality holds, and in particular (8) holds.
Remark 4.3. Note that in the above formulation each is an element of (ℝ ). Instead one could take to be an element of ( (ℝ)) , and the same statements with analogous proofs would still hold. This alternative formulation has a more transparent financial interpretation as it corresponds only to marginal laws of terminal values of each stock price as opposed to the full distribution; see also Lim (2016) for a related discussion.

Dynamic programming principle onÊ
ldan (2016) and Cox and Källblad (2017) studied the Skorokhod embedding and martingale optimal transport problems in continuous time using measure-valued martingales. This point of view enables one to obtain the dynamic programming principle with marginal constraint because the terminal constraint is transformed into the initial constraint. We adopt this perspective which proves to be very useful.

Proof.
(a) Let ⊂ ℝ and recall that =̂( )Q-a.s. Then, we have where the second equality holds asQ is terminating and the third one follows by the definition of the MVM measure in Definition 4.4 as well as remark 2.2 of Cox and Källblad (2017). Hence, the first assertion is proven.
(b) Let ≤ , ≤ and be a convex function. Then, where the first and the last equalities follow by (a), the penultimate is due to the consistency ofQ, and the inequality follows by conditional Jensen's inequality. □ Recall the set of martingale measures in Definition 2.13. The following lemma shows how to build the map and that (Ω,̂, , , , ) is a dynamic extension of (Ω, ,  , , , ). Proof.
(a) The process =̂( ) is a (Q,̂)-martingale asQ is an MVM measure. To prove that is a (Q,̂)-martingale for any ∈ Λ, it is enough to show that for any ≤ and ∈ Λ 0 one haŝ ℚ [ (̂( ))| ] =̂( ) for any < . The latter holds becausê where the first equality follows by consistency ofQ, the second holds asQ is terminating, and the last one holds becauseQ is an MVM measure. Let us define a family of operators , and so forth, as in Example 2.12, ,̂∈Υ, and then the extension as well as 0 on the enlarged space as in Section 2.2. We then have the following theorem.
In particular the pricing-hedging duality (24) holds in this martingale optimal transport context for all functionals Φ ∶ Ω → ℝ , which are upper semicontinuous and bounded from above.
Proof. Notice that the pricing-hedging duality on Ω holds by Theorem 4.2. Then, by Corollary 2.19, it is enough to establish the dynamic programming principle onΩ to prove the pricing-hedging duality (24). Using exactly the same arguments as in (4.12) of Bouchard and Nutz (2015), to establish the dynamic programming principle onΩ, it is enough to argue that is such that } is analytic.
To prove the above analyticity property, we first observe that

PROOFS FOR SECTION 2
We recall that Section 2 stays in a context with an abstract space (Ω,  ) equipped with an underlying process and a family  of probability measures.  denotes the collection of all measures ℚ dominated by some ℙ ∈  and such that is a ℚ-martingale, and sup ℚ∈ ℚ [ ⋅ ] admits a dynamic programming representation by  (see (9)), from which one defines the family of operators in (10). A first enlarged space Ω ∶= Ω × {1, … , } is introduced in Section 2.2 to reduce an American option to a European option, and a dynamic extensionΩ of Ω is defined in Definition 2.13 to introduce a fictitious market allowing dynamic trading of options.
Proof. In the context of Example 2.12, the family ( ) take the following form: To see that (13) holds, it is insightful to rewrite  0 in a slightly different way, as 0 below. Recall that  is the universal completion of  , where the latter is countably generated. Let where [ ]  − is defined as in (18). Next, for Φ ∈ Υ, let us introduce the operators Note that the regular conditional probabilities of any ℚ ∈  with respect to   Proof. The proof follows by exactly the same arguments as in Lemma A.2 of Bouchard and Nutz (2015), using the discrete time local martingale characterization in their Lemma A.1.
By Lemma 6.2, we easily obtain the weak duality for all upper semianalytic Φ The following lemma shows that, for a fixed Φ, we can restrict to martingale measures satisfying a further integrability constraint.
Lemma 6.3. Let Φ be upper semianalytic, ℚ ∈  0 and ∶ Ω → [1, ∞) be such that |Φ( , )| ≤ ( ) for all = ( , ) ∈ Ω. Then  ,ℚ ≠ ∅ and Proof. First, by Lemma A.3 of Bouchard and Nutz (2015), there exists a probability measure ℙ * , equivalent to ℚ on (Ω,  ), such that ℙ * [ ] < ∞. On the filtered probability space (Ω,  , , ℙ * ), one defines  * as the collection of all probability measures ℚ then by the classical arguments for the dominated discrete time market, such as Kabanov (2008) and Kabanov and Stricker (2001), see also lemma A.3 of Bouchard and Nutz (2015), one can easily obtain the inequality which concludes the proof. □ Using theorem 2.2 of Bouchard and Nutz (2015), which is stated for a general abstract space (Ω,  ), one directly obtains a closeness result for the set of all payoffs, which can be superreplicated from initial capital = 0, in our context. Let us denote by  0 + the set of all positive random variables on Ω, and define  ∶= (2015)). Let Φ be upper semianalytic and assume that NA() holds. Then the set  is closed in the following sense. Whenever ( ) ≥1 ⊂  and is a random variable such that → , -q.s., then ∈ .

Proof of Theorem 3.2: the case = , equivalently = ∅
For each 1 ≤ ≤ ≤ , we introduce a map from Ω to Ω (resp. Note that  − is the smallest -field on Ω generated by [ ⋅ ] ∶ Ω → Ω ; equivalently, an  − -measurable random variable defined on Ω can be identified as a Borel measurable function on Ω . The process is naturally defined on the restricted spaces Ω and Ω . We next recall the notion of local no-arbitrage condition NA( ( )) introduced at the beginning of section 4.2 in Bouchard and Nutz (2015). Given a fixed ∈ Ω , we can consider Δ +1 ( , ⋅) ∶= +1 ( , ⋅) − ( ) as a random variable on Ω 1 , which determines a one-period market on (Ω 1 , (Ω 1 )) endowed with a class  ( ) of probability measures. Then, NA( ( )) denotes the corresponding no-arbitrage condition in this one-period market, that is, NA( ( )) holds if for all ∈ ℝ , Lemma 6.5. In the context of Section 3, let ∶ Ω +1 → ℝ be upper semianalytic. Then,  ( ) ∶ Ω → ℝ is still upper semianalytic. Moreover, there exist two universally measurable functions ( 1 , 2 ) ∶ Ω → ℝ × ℝ such that Proof. Notice that 1 ∨ 2 is upper semianalytic whenever 1 and 2 are both upper semianalytic. Using the definition of  , the above lemma follows by applying lemma 4.10 of Bouchard and Nutz (2015) for every fixed .
Proof of Theorem 3.2 (the case = 0). First, one has the weak duality as in (45) Next, for the inverse inequality, we can assume, without loss of generality, that Φ is bounded from above. Indeed, by Lemma 6.4, one has lim →∞ (Φ ∧ ) = (Φ); see also the proof of theorem 3.4 of Bouchard and Nutz (2015). Besides, the approximation sup ℚ∈ is an easy consequence of the monotone convergence theorem.
As ∶= { ∶ NA( ( )) fails} is -polar by theorem 4.5 of Bouchard and Nutz (2015), it follows that, with To conclude, it is enough to notice that the above is an optimal dual strategy for the case of Φ being bounded from above. The existence of the optimal dual strategy for general Φ is then a consequence of Lemma 6.4.

Proof of Theorem 3.2: The case ≥ , equivalently ≠ ∅
We will adapt the arguments in section 5 of Bouchard and Nutz (2015) to prove Theorem 3.2 in the context with finitely many options ≥ 1.
For technical reasons, we introduce which depends only on , and Moreover, in view of Lemma 6.3, one has Proof of Theorem 3.2 (the case ≥ 1). The existence of some ℚ ∈  is an easy consequence of theorem 5.1 of Bouchard and Nutz (2015) under NA(). Moreover, similarly to Bouchard and Nutz (2015), there exists an optimal dual strategy by Lemma 6.4. Let us now focus on the duality results. First, the duality (Φ) = sup ℚ∈ ℚ [Φ] in (34) has already been proved for the case = 0. We will use an inductive argument. Suppose that the duality (34) holds true for the case with ≥ 0, We aim to prove the duality with + 1 options where the additional option has a Borel measurable payoff function ≡ +1 such that | | ≤ , and has an initial price 0 = 0. By the weak duality in (45) and Lemma 6.3, the "≥" side of the inequality holds true, so we will focus on the "≤" side of the inequality, that is, If is replicable by some semistatic strategy with underlying and options ( 1 , … , ) in the sense that ∃ ∈ , ℎ ∈ ℝ , such that = ( • ) + ℎ , -q.s. (or equivalently, ∃ ∈ , ℎ ∈ ℝ , such that = ( • ) + ℎ , -q.s.), then the problem is reduced to the case with options and the result is trivial. Let us assume that is not replicable, and we claim that there exists a sequence (ℚ ) ≥1 ⊂  such that Next, denote by ( ) the minimum superhedging cost of the European option using and ( 1 , … , ), that is, As is not replicable, by the second fundamental theorem in Theorem 5.1.(c) of Bouchard and Nutz (2015), we have that ℚ  → ℚ [ ] is not constant on  . Then, under the no-arbitrage condition, one Thus, there exists some ℚ + ∈  , such that 0 < ℚ + [ ] < ( ). With the same argument on − , we can find another ℚ − ∈  such that Then, one can choose an appropriate sequence of weights ( − , 0 , + ) ∈ ℝ 3 + , such that − + 0 + + = 1, ± → 0, , and we hence have the inequality (48).
It is enough to prove the claim (49), for which we suppose without loss of generality that ( , ) (Φ) = 0. Assuming that (49) fails, one then has By the convexity of the above set and the separation argument, there exists ( , ) ∈ ℝ 2 with |( , )| = 1, such that The strict inequality ( , ) is nonempty under the NA() assumption in the case of + 1 options.
(b) Next, we claim that the graph set is analytic. Then using the (analytic) measurable projection theorem; see, for example, proposition 4.47 of Bertsekas and Shreve (2007), (̂) is upper semianalytic whenever̂is upper semianalytic. Further, the second equality in (35) is just a classical dynamic programming principle result, which follows by the measurable selection arguments as in Dellacherie (1985) or Bertsekas and Shreve (2007).

PROOFS FOR SECTION 4
We finally complete here the proof of Theorem 4.2, which concerns the pricing-hedging duality in the martingale optimal transport problem setup. Recall that, in this setup, Ω ∶= Ω × {1, … , }, with Ω ∶= ℝ × , and  the collection of all Borel probability measures on Ω.
A first idea of how to prove Theorem 4.2 could be the following two-step argument as in Guo et al. (2016). First, under the condition that Φ is bounded from above and upper semicontinuous, one could prove that is concave and upper semicontinuous, where we equip ((ℝ ) ) with a Wasserstein type topology. Second, using the Fenchel-Moreau theorem, it follows that Solving the maximization problem (52), by using Theorem 3.2, concludes the proof of Theorem 4.2. However, in the following, we will provide another proof based on an approximation argument. For simplicity, we suppose that 0 = { }, where the same arguments work for more general 0 . In preparation, let us provide a technical lemma. In the context of the martingale optimal transport problem, we introduce a sequence of payoff functions ( ) ≥1 by where ∶ ℝ → ℝ is Lipschitz and ( ) ≥1 is dense in the space of all Lipschitz functions on ℝ under the uniform convergence topology, and moreover, it contains all functions of the form ( − ) + , (− − ) + for = 1, … , and ≥ 1, where ( ) ≥1 ⊂ ℝ is a sequence such that → ∞. Notice that has finite first order and hence the are all finite constants.
Next, let us introduce an approximate dual problem by (b) The sequence ( , ℚ ) ≥1 is uniformly integrable, and any accumulation point of (ℚ ) ≥1 belongs to  . Proof.
(d) To conclude the proof, it is enough to show that the martingale property is preserved for the limiting measure ℚ 0 . By extracting a subsequence, we assume that ℚ → ℚ 0 weakly. First, for any = 1, … , , = 1, … , and > 0, one has It follows that  is included in the vector space of all bounded  1 -measurable random variables for which ℚ 0 [ ( 2 − 1 )] = 0. An application of the monotone class theorem (see, e.g., theorem I.8 of Protter, 2005) yields ℚ 0 [ ( 2 − 1 )] = 0, for all bounded  1 -measurable random variables , which is equivalent to being a ℚ 0 -martingale, and concludes the proof.
Proof of Theorem 4.2. We notice that by Theorem 3.2, sup ℚ∈ , Let (ℚ ) ≥1 be a sequence of probability measures such that ℚ ∈  , for each ≥ 1 and and we hence conclude the proof by the weak duality (45).

CONFLICT OF INTEREST
The authors have declared no conflict of interest.